Method for design of pricing schedules in utility contracts

ABSTRACT

A provider of standardized services is provided with guidance on the design of pricing structures for contracts regulating the provision of a commodity good between a supplier and a customer. These are contracts characterized by long duration and dedicated infrastructure. The provision of the commodity good is variable over time, and the rate of provisioning is continuously monitored. Examples are kilowatt hours in the case of electric energy and megabytes/second in the case of Web hosting.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention generally relates to the design ofcontracts for outsourced information services having similarities tocontracts that are commonly adopted by suppliers of utility servicesand, more particularly, to the design of contracts for outsourcedservices provided by the information technology (IT) industry whereincustomers are charged according to their actual resource usage duringthe term of the contract.

[0003] 2. Background Description

[0004] Information services and utility services share one essentialfeature—the demand for such services varies over time. A Web hostingprovider, a data storage facility, or a regional electric power provideroffer contracts to corporate customers in which the provisioning oftheir service is allowed to vary during the contract interval. In thesecontracts, a central role is played by the pricing schedule, whichdetermines the service charge based on the observed demand. Severalconsiderations enter into the design of an effective pricing schedule.For example, the provider might take into account the differences inpreferences among customers to design a nonlinear scheme that maximizesprofits (R. B. Wilson, Nonlinear Pricing, Oxford University Press, NY,1993). A consideration of a different nature is the risk faced by theprovider. If the final charge is nearly independent of the usage, as ina fixed charge price, a customer with low demand might not find thecontract attractive and walk away. On the other hand, if the charge isstrongly dependent on the usage, the provider might not be able torecover its costs in the case of a customer with low demand. The pricingdilemma faced by the provider is linked to the costs the provider isincurring before the customer demand is observed.

[0005] Pricing for utility contracts has been explored by S. Oren, S.Smith and R. Wilson in “Capacity pricing”, Econometrica, 53(3):545-566(1985), in the context of single-stage contracts. In their analysis,customers purchase in advance a consumption profile from a monopolist.J. Panzar and D. Sibley in “Public utility pricing under risk: the caseof self-rationing”, The American Economic Review, 68(5):888-895 (1978),propose a two-stage setting. In their analysis, the customer purchases apeak rate in the first stage and is allowed to choose a consumptionlevel during the second stage, provided that the consumption rate doesnot exceed the peak rate. The resulting equilibrium is not necessarilyPareto-optimal.

[0006] When considered as a newsvendor problem, the model can beinterpreted as an optimal ordering problem in two stages, in whichadditional information is received before the second order. In thisframework, the literature on channel coordination is vast and growing.M. Fisher and A. Raman in “Reducing the cost of uncertainty throughaccurate response to early sales”, Operations Research, 44(1):87-99(1996), model the problem as a two stage production decision process, inwhich additional information for early sales is taken into account whensetting production quantities in the second stage. G. D. Eppen and A. V.Iyer in “Backup agreements in fashion buying—the value of upstreamflexibility”, Management Science, 43(11):1469-1484 (1997), also considera two-stage setting, under different contractual agreements. L.Weatherford and P. Pfeifer in “The economic value of using advancebooking of orders”, Omega, 22(1):405-411 (1994), analyze theinformational advantage of advanced book-to-order in the case ofnormally distributed demands in stages one and two with knowncorrelation. A. V. Iyer and M. E. Bergen in “Quick response inmanufacturer-retailer channels”, Management Science, 43(4):559-570(1997), study the benefits of multi-stage transactions between aretailer and a supplier, achieved via Bayesian updating of thesupplier's beliefs. A taxonomy of scenarios in which the interestedparties have asymmetric information is also presented by A. H.-L. Lauand H.-S. Lau in “Some two-echelon style-goods inventory models withasymmetric market information”, European J Oper. Res., 134:29-42 (2001),under specific demand assumptions.

[0007] The strategic analysis of centralized and decentralized behaviorin inventory management is relatively recent. The articles of H. Lee andS. Whang, “Decentralized multi-echelon supply chains: Incentives andinformation”, Management Science, 45(5):633-640 (1999), and G. P. Cachonand P. H. Zipkin, “Competitive and cooperative inventory policies in atwo-stage supply chain”, Management Science, 45(7):936-953 (1999), showhow channel coordination may be achieved through a variety ofmechanisms, such as linear transfer, and penalties rewards contingent onthe observed demand. Finally, option mechanisms in inventory managementhave been proposed recently D. Shi, R. Daniels and W. Grey in “The Roleof Options in Managing Supply Chain Risks”, IBM Research Report RC 21960(2001).

[0008] Recently, the need for standardized information services hasinspired the deployment of a new class of outsourcing services in theinformation technology (IT) industry. In these new offerings, customersare charged according to their actual resource usage during the contractduration, This represents a radical departure from past outsourcingcontracts. The flexibility is desirable for the customer in a sectorwith high fixed costs, low marginal costs, and high depreciation ratesfor equipment.

[0009] Outsourcing contracts exhibit several distinctive features.First, the transactions are not directly generated by the customer, butby a large number of agents who have some relationship with him. Forexample, these agents can be the employees of a company, or thesubscribers to an online service. This market structure has an importantimplication for the type of the contract—the arrival process oftransactions is exogenous; i.e., its features are independent of thecontractual obligations between customer and -provider. A second featurecommon to such contracts is that they are exclusive. The customer agreesto receive the service by only one provider for the contract duration.Finally, resale of the service is prohibited.

[0010] In the basic service setting, a customer signs a contract offixed duration with the service provider. The contract specifies one ormore service unit (SU). The SU is defined as a transaction of a certaintype initiated by the customer and processed by the provider's servicecenter. The SU depends on the context. For example, in the case of Webcaching services, a possible unit would be a hypertext transfer protocol(http) GET request, while in the case of a managed storage service, theSU would be a megabyte (MB) of data transferred between customer andprovider. The SU rate is continuously monitored by the provider. Thefinal charge to the customer is contingent on the realization of theservice rate curve. Within the framework outlined above, the pricingscheme adopted by the provider constitutes the core of the contract.

SUMMARY OF THE INVENTION

[0011] It is therefore an object of the present invention to provide asolution to the pricing dilemma faced by the provider of informationservices.

[0012] According to the invention, “computing utilities” deliverprocesses running on a shared infrastructure, with standardized servicemetrics, and with prices that reflect the amount of service received.The initial capacity investment decision is critical to the success of anew offering. The problem of capacity allocation under a linear pricingcontract resembles that of a newsvendor problem. A new pricing scheduleis introduced in which, at the beginning of the contract, the customercan set a load threshold, below which the customer is charged adiscounted unit price. If the customer has private information on his orher load characteristics, the invention attains full informationrevelation, and results in the highest possible utilitarian welfare forthe system. The contract parameters can be computed based on the costparameters of the problem, such as unit capacity costs and penaltycosts. In addition, there is a family of price schedules that results inallocations for provider and customer that are a Pareto improvement overthe standard schedule.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The foregoing and other objects, aspects and advantages will bebetter understood from the following detailed description of a preferredembodiment of the invention with reference to the drawings, in which:

[0014]FIG. 1 is a block diagram of an exemplary system showing aninformation source provider connected through the Internet to aplurality of customers;

[0015]FIG. 2 is a graph showing the structure of a flexible discountcontract;

[0016]FIG. 2A is a time line showing a sequence of events in thedecision process;

[0017]FIG. 3 is a graph showing the expected allocating under the linearand flexible discount pricing;

[0018]FIG. 4 is a graph showing welfare allocations under the linear andflexible discounts contracts;

[0019]FIG. 5 is a flowchart showing the basic process according to theinvention;

[0020]FIG. 6 is a flowchart showing the logic of the monitoring process;and

[0021]FIG. 7 is a flowchart showing the logic of the computationprocess.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT OF THE INVENTION

[0022] Referring now to the drawings, and more particularly to FIG. 1,there is shown a source provider 10 connected through the Internet 12 toa plurality of customers 14 to 16. The problem solved by this inventionis the pricing of the services provided by the source provider 10 to theseveral customers 14 to 16. More specifically, the invention provides apricing schedule in which, at the beginning of the contract, thecustomer can set a load threshold, below which he or she is charged adiscounted unit price. The contract parameters can be computed based onsuch cost parameters as unit capacity and penalty costs.

[0023] The service provider 10 in the illustrated embodiment of FIG. 1comprises a server 111 which is connected to the customers 14 to 16through the Internet 12. The server 111 provides data to a load monitor112 which monitors the loads of each of the individual customers 14 to16. The monitored load time series as monitored by the load monitor 112are stored in a repository 113. A pricing and billing component 114 ofthe service provider 10 accesses the load time series stored in therepository 113 and computes bills to each of the individual customers 14to 16.

[0024] In the following description of the invention, a contracttemplate which subsumes some contracts adopted in utility sectors,notably in the energy wholesaler/retailer and the IT outsourcingsectors, is analyzed. In this contract, the customer is charged based onthe number of SU received by the customer during the contract duration.There are two objectives. First, there is provided a rationale for theexistence of contracts that are popular among practitioners, but havereceived little attention among researchers. The contract is amenable tothree interpretations. In the first, the contract can be viewed as anonlinear pricing schedule in which the customer nominates the thresholdfor quantity discount. In the second, the contract is a bundle ofoptions contingent on the observed customer load, in addition to “spot”contracts. In the third, the contract requires the customer to commit toa certain threshold, for which he pays ex ante, but gives him rebatesfor non-used capacity and SU above his committed capacity, but at apremium price. During the analysis, there is established a link betweenthe flexible discount contract and the newsvendor model, that is oftenused in the supply-chain literature to model the relationship betweenthe manufacturer of a perishable good and a retailer.

[0025] The second objective is to provide guidelines for the design ofbetter contracts; i.e., contracts that achieve a higher social optimumand/or a higher rent for the provider. The provider can improve upon thebasic usage-based contract by eliciting private information on thecustomer's demand profile. In particular, through a correct choice ofthe contracts parameters, the provider receives a rent that isarbitrarily close to the highest possible rent. The result does not makeany assumption, neither on the probability distribution of demand nor onthe probability distribution of customer profiles. When interpreted inthe context of a newsvendor problem, the results show that, under theflexible discount contract, retailer and wholesaler achieve maximumchannel coordination.

Model Formulation

[0026] The contract time interval is divided into N sampling intervalsof equal length. For each sampling interval n=1, . . . , N, the providermeasures the number X_(n), of Service Units (SU) provisioned to thecustomer. The cost structure of the provider is divided in long-run andshort-run capacity costs. Before the starting date of the contract, theprovider chooses his resource capacity q, where q is defined as themaximum number of SUs that can be served during a sampling interval. Letthe unit cost of this capacity be c per sampling interval. If the demandduring a sampling interval exceeds the capacity q, the provider canserve it by incurring a unit cost equal to c′, which is assumed to bestrictly greater than c. This peak service is amenable to differentinterpretations. In some contexts, such as in electric power generationplants, the provider might own “spinning” generation units, which canprovide short-run capacity, at higher marginal costs. In differentcontexts, such as Web hosting, idle servers might be dynamicallyreconfigured to serve the excess demand. Finally, if no physicalcapacity is available, c′ models the financial reimbursement paid by theprovider in the case of denial of service, or might be a proxy forlong-term losses due to reduced customer good will. If some units of thelong-run capacity allocated to the provider are not used during asampling interval, they can be salvaged during that interval, forexample by diverting them for a different task. Let the salvage revenuebe s per SU. We will use the shorthand mathematical notationx⁺=max{x,0}, xΛy=min{x,y}, and

{x≦y}=1 if x≦y, and 0 otherwise.

[0027] LINEAR PRICING: In the simplest form of a usage-based contract,the provider charges a unit price p per SU. It is assumed that theprovider is a price-taker, so that p is not a decision variable. Theprofit of the provider is then equal to $\begin{matrix}{V_{N} = {{p{\sum\limits_{n = 1}^{N}X_{n}}} + {s{\sum\limits_{n = 1}^{N}\left( {q - X_{n}} \right)^{+}}} - {c^{\prime}{\sum\limits_{n = 1}^{N}\left( {X_{n} - q} \right)^{+}}} - {Ncq}}} & (1)\end{matrix}$

[0028] In addition to the contract introduced above, two-stage contractsare commonly used. In these contracts the customer selects a pricingschedule from a menu before the demand is observed (ex ante) and pays afee that depends on the contract chosen. At the end of the contractinterval the customer pays the provider a rent contingent on theobserved demand and on the pricing schedule. Attention is concentratedon the following contract.

[0029] FLEXIBLE DISCOUNT: The customer reserves ex ante a discountthreshold r, for which the customer pays a unit price Np₀. During asampling interval the customer pays a discounted unit price p₁ if loaddoes not exceed r, and pays the full price p if the load exceeds r. Analternative interpretation of the pricing schedule is the following:before the customer observes demand, the customer buys r call options ata price p that gives the customer the right to buy a SU at a unit pricep, during each sampling interval. During each sampling interval thecustomer exercises his options. Another possible interpretation is ofthe contract is as committed capacity with rebates and penalties: beforehe observes demand, the customer buy a capacity r at unit priceN(p₀+p₁). During each sampling interval, the customer receives a rebateequal to p₁ for each SU of his allotted capacity that has not been used,and pays an unit price p for each SU that the customer has used abovethe customer's allotted capacity. The final profit is then equal to$\begin{matrix}{V_{N} = {{p_{0}r} + {p_{1}{\sum\limits_{n = 1}^{N}\left( {X_{n}\bigwedge r} \right)^{+}}} + {p{\sum\limits_{n = 1}^{N}\left( {X_{n} - r} \right)^{+}}} + {s{\sum\limits_{n = 1}^{N}\left( {q - X_{n}} \right)^{+}}} - {c^{\prime}{\sum\limits_{n = 1}^{N}\left( {X_{n} - q} \right)^{+}}} - {Ncq}}} & (2)\end{matrix}$

[0030] The flexible discount scheme is illustrated in FIG. 2.

[0031] Some remarks are in order. In this analysis, the unit price p isshared among pricing schedules. This is considered the reference priceper SU. Also, it is noted that when the number of measurements N islarge, the pricing formula can be approximated by a simpler, asymptoticexpression. Let${\overset{\_}{V}}_{N}:={{\frac{\pi_{N}}{N}\quad \text{and}\quad {F_{N}(x)}} = {\frac{1}{N}{\sum\limits_{n = 1}^{N}{1{\left\{ {X_{n} \leq x} \right\}.}}}}}$

[0032] THEOREM 1: If the load process {X_(n), n≧0} is stationary,integrable and ergodic, then the limits V=lim {overscore (V)}_(N), andF(•) exist, and

V=p ₀ r+p ₁ E(DΛr)+pE(D−r)⁺ +sE(q−D)⁺ −c′E(D−q)⁺ −cq  (3)

[0033] where D is a rv with CDF equal to F(•).

[0034] PROOF OF THEOREM 1: Birkhoff's ergodic theorem (R. Durrett,Probability: theory and examples, Duxbury Press, Belmont, Calif., 2^(nd)Ed., 1996) states that, for any measurable function h(•), we have lim${{N^{- 1}{\sum\limits_{n = 1}^{N}{f\left( X_{n} \right)}}} = {E\left\lbrack {h(X)} \right\rbrack}},$

[0035] where X is a random variable with cumulative distributionfunction given by${F(x)} = {N^{- 1}{\sum\limits_{n = 1}^{N}{1{\left\{ {X_{n} \leq x} \right\}.}}}}$

[0036] Applying this result to each element of the right hand side ofEquation (1) the result follows.

[0037] If restricted to the linear pricing contract, Formula (3) becomes

V=pE(D−r)⁺ +sE(q−D)⁺ −c′E(D−q)⁺ −cq.

[0038] The above formula bears a close resemblance with the newsvendormodel. In the folk version of the problem, a wholesaler commits tosatisfy the demand for a certain product of a retailer, and must decidein advance which quantity to order before the retailer's demand isobserved. After the ordering decision is made, demand is revealed. Ifdemand is lower than supply, the unsold product can be salvaged. On theother hand, if the wholesaler receives an order from the retailer thatexceeds his available supply, he meets the demand by purchasingadditional product at a premium price. In this notation, D representsthe random demand of a product; q is the wholesaler's advance order atcost c; unit price paid by the retailer is p; unit salvage revenue is s;while cost for late orders is c′. It is assumed that c′>p>c>s. Thenewsvendor model and its variants have been used to model inventorydecision problems in which the product has a short lifetime. The profitcan be expressed as

πr(q, D)=pD+s(q−D)⁺ −c′(D−q)⁺ −cq.

[0039] The optimization problem has a unique solution$\hat{q} = {{F_{D}^{- 1}\left( \frac{c^{\prime} - c}{c^{\prime} - s} \right)}.}$

[0040] The value $\hat{f} = \frac{c^{\prime} - c}{c^{\prime} - s}$

[0041] is called the critical fractile.

The Role of Commitment in Outsourcing Contracts

[0042] In order to increase expected profit, the provider can attempt togain additional information on the customer's demand distribution. Tomake this statement precise, let us assume that the customer has a typeθεΘ; the type is a vector that captures the heterogeneity of thecustomer population, and takes values in a subset of a euclidean space.The type contains the sufficient statistics of customer's demand Xn. Forexample, consider the case where the Xn are independent, identicallydistributed normal random variables. The type would be theta=(mu,sigma), i.e., the mean and standard deviation associated to the normaldistribution. As a consequence, the type determines the statisticalproperties of the customer demand; i.e., the cumulative distributionfunction of demand for a customer of type θ can be written asF_(D|θ)(X|θ). We assume that the functional form of F_(D|θ)(•|•) isknown to both provider and customer, and, for the sake of simplicity, weshall assume that for each θε|Θ, F_(D|θ)(•) be a continuous function.The customer has knowledge of his own type, while the provider has aprior probability measure P_(θ)on Θ for the customer type.

[0043] Under the linear pricing contract, the provider's optimalexpected profit is given by $\begin{matrix}\begin{matrix}{V_{1} = {\max\limits_{q}{E\left( {\pi \left( {q,D} \right)} \right)}}} \\{= {\max\limits_{q}\left( {\Pi \left( {q,\theta} \right)} \right)}}\end{matrix} & (5)\end{matrix}$

[0044] where II(q, θ)=E(π(q, D)|θ), the expected profit when producedquantity is q and customer's type is θ. Suppose that some additionalinformation F the distribution of types is available to the providerbefore he or she has to decide q. Intuitively, F is the knowledge that θbelongs to a subset of Θ. The optimal expected profit conditional on Fbecomes${E\left( {\max\limits_{q}{E\left( {\Pi \left( {q,\theta} \right)} \middle| \mathcal{F} \right)}} \right)}{q.}$

[0045] Let h(π, F) be the value of information (VOI) associated to F,defined as the difference between optimal profit in the presence ofinformation F and optimal profit without additional information.$\begin{matrix}{\left. {{h\left( {\pi,\mathcal{F}} \right)} = {E\left( {\max\limits_{q}{\Pi \left( {q,\theta} \right)}} \middle| \mathcal{F} \right)}} \right) - {\max\limits_{q}{E\left( {\Pi \left( {q,\theta} \right)} \right)}}} \\{\left. {= {{E\left( {\max\limits_{q}{E\left( {\Pi \left( {q,\theta} \right)} \middle| \mathcal{F} \right)}} \right)} - {\max\limits_{q}{E\left( {\Pi \left( {q,\theta} \right)} \middle| \mathcal{F} \right)}}}} \right).}\end{matrix}$

[0046] It is a well-known result that h(π, F) is nonnegative (see M.Avriel and A. Williams, “The value of information and stochasticprogramming”, Operations Research, 18(5):947-954, 1970). The VOI ismaximized when the type of the customer is known exactly (I. H. LaValle, “On cash equivalents and information evaluation in decisionsunder uncertainty: Part I: Basic Theory”, Journal of the AmericanStatistical Association, 63(321):252-276 1968). For all F,$\begin{matrix}\begin{matrix}{{E\left( {\max\limits_{q}\left( {\Pi \left( {q,\theta} \right)} \middle| \mathcal{F} \right)} \right)} \leq {E\left( {\max\limits_{q}{E\left( {\pi \left( {q,D} \right)} \middle| \theta \right)}} \right)}} \\{= {E\left( {\max\limits_{q}{\Pi \left( {q,\theta} \right)}} \right)}} \\{= {E\left( {\Pi \left( {{\hat{q}(\theta)},\theta} \right)} \right)}} \\{{= V_{FB}},}\end{matrix} & (6)\end{matrix}$

[0047] where V_(FB) is the first-best solution and

{circumflex over (q)}(θ)=F _(D|θ) ⁻¹({circumflex over (f)})  (7)

[0048] is the optimal solution of the standard newsvendor problem whenthe type is known.

[0049] Based on the above observation, it is desirable for the providerto obtain additional information on the customer's type in order toincrease the expected profit. There are several ways to obtainadditional information about the customer's type. For example,interviews, market surveys and information contained in historical dataof the customer's demand can provide useful information about hiscumulative distribution function. There are several drawbacks tofollowing this approach. The first one is that market research isexpensive and time-consuming. Moreover, the information contained insuch research might be unreliable. As an alternative, the provider canattempt to elicit information within the terms and communicationchannels established by the contract. The rationale behind ourformulation of two-stage contracts is that the first stage serves adevice to elicit the information relative to the customer's type that isrelevant for capacity planning. Consider the flexible discount contract.The sequence of events is illustrated in FIG. 2A. In the first stage 81of the decision process, the provider chooses parameters p₀,p₁ andoffers the contracts. In the second stage 82 of the decision process,the customer chooses the number of contracts r. In the third stage 83,the provider sets the production level q using the availableinformation. It is assumed that both provider and customer arerisk-neutral, and that they maximize the net present value of theirmonetary transfers. For simplicity, the interest rate is set to zero.The main result can be formally stated as follows.

[0050] THEOREM 2: For any ε>0, let

p ₁ε(p−ε/E({circumflex over (q)}(θ)), p)  (8)

[0051] $\begin{matrix}{p_{0} = {\frac{c - s}{c^{\prime} - s}{\left( {p - p_{1}} \right).}}} & (9)\end{matrix}$

[0052] Then, the provider expected profit V(p₀,p₁) is such that

Vε(V _(FB) −ε, V _(FB)).

[0053] Furthermore, the optimal production level q* is given by r*, thenumber of contracts purchased by the customer in the second stage, andis independent of the choice of p₁, as long as p₀ satisfies Equation(9).

[0054] PROOF OF THEOREM 2: The contract can be formulated as asequential game in five stages, as shown in FIG. 4. In the first stage91, Nature chooses the customer's type according to a probabilitymeasure P_(θ) defined on the space Θ, which we assume to be the subsetof a Euclidean space. In the second stage 92, the provider choose thevalues of p₀, p₁. In the third stage 93, the customer chooses the numberof committed units r that minimize his expected cost. In the fourthstage 94, the provider chooses a production quantity q that maximize hisexpected profit, based on the available information. In the final stage95, Nature chooses the state of the world W from a space Q. Demand is afunction of both the observed state of the world and the customer'stype, and we write D(ω,θ). We can express the provider's profit π′(p₀,p₁, r, q, ω,ψ) as follows: $\begin{matrix}\begin{matrix}{\pi^{\prime} = {{p_{0}r} + {p_{1}\left( {{D\left( {\omega,\theta} \right)}\bigwedge r} \right)} + {p\left( {{D\left( {\omega,\theta} \right)} - r} \right)} +}} \\{{{s\left( {q - {D\left( {\omega,\theta} \right)}} \right)}^{+} - {cq} - {c^{\prime}\left( {{D\left( {\omega,\theta} \right)} - q} \right)}^{+}}} \\{\left. {= {{p_{0}r} + {\left( {p_{1} - p} \right){{D\left( {\omega,\theta} \right)}\bigwedge r}}}} \right) + {{pD}\left( {\omega,\theta} \right)} +} \\{{{s\left( {q - {D\left( {\omega,\theta} \right)}} \right)}^{+} - {c^{\prime}\left( {D - {q\left( {\omega,\theta} \right)}} \right)}^{+} -}} \\{{c\left( {q\left( {\omega,\theta} \right)} \right)}}\end{matrix} & (10) \\{= {{p_{0}r} + {\left( {p_{1} - p} \right)\left( {{D\left( {\omega,\theta} \right)}\bigwedge r} \right)} + {\pi \left( {q,{D\left( {\omega,\theta} \right)}} \right)}}} & (11)\end{matrix}$

[0055] where π(q, D) is defined in Equation (4). The cost incurred bythe customer is given by $\begin{matrix}\begin{matrix}{{\kappa \left( {{p0},{p1},r,\omega,\theta} \right)} = {{p_{0}r} + {p_{1}\left( {{D\left( {\omega,\theta} \right)}\bigwedge r} \right)} + {p\left( {{D\left( {\omega,\theta} \right)} - r} \right)}^{+}}} \\\left. {= {{p_{0}r} + {\left( {p_{1} - p} \right)\left( {{D\left( {\omega,\theta} \right)}\bigwedge r} \right)} + {p\quad {D\left( {\omega,\theta} \right)}}}} \right)\end{matrix} & (12)\end{matrix}$

[0056] The last stage of the game is a lottery with expected payoffsequal to

II(p ₀ , p ₁ , r, q, θ)=E(π′(p ₀)p ₁ , r, q, ω, θ)|θ)

K(p ₀ , p ₁ , r, θ)=E(κ(p ₀ , p ₁ , r, ω, θ)|θ)

[0057] The game can be therefore reduced to a four-stage game, whoseextensive form representation is shown in FIG. 4.

[0058] The concept of Weak Perfect Bayesian Equilibrium (A. Mas-Colell,M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford UniversityPress, 1995) is employed to determine the equilibrium strategies of thisgame. This is equivalent to finding beliefs on the customer's type thatare consistent; i.e., that are derived using Bayes' rule wheneverpossible and to finding strategies for both the provider and thecustomer that are sequentially rational given the set of beliefs. In thecontext of this specific game, these requirements take a simple form.Consider the subgame comprised by stages three and four in FIG. 4. Atype-θ customer's equilibrium strategy is described by a probabilitydistribution P_(c)(r, θ) on the capacity commitment r. After theprovider observes the commitment r, the provider's equilibrium strategyis described by a probability distribution P_(p)(q, r) on the capacityreservation q. The customer chooses a capacity r with positiveprobability only if this value minimizes the expected cost, based on theprovider's strategy: $\begin{matrix}{r \in {\arg \quad {\min\limits_{r}{\int{{K\left( {p_{0},p_{1},r,\theta} \right)}{{P_{p}\left( {q,r} \right)}}}}}}} & (14)\end{matrix}$

[0059] Similarly, q is in the support of P_(p)(•, r) only if this valuemaximizes the expected profit, based on the customer's strategy:$\begin{matrix}{q \in {\arg \quad {\max\limits_{q}{\int{{\Pi^{\prime}\left( {p_{0},p_{1},r,q,\theta} \right)}{{F_{\theta|r}(\theta)}}}}}}} & (15)\end{matrix}$

[0060] where the provider updates his beliefs of the distribution of θaccording to Bayes' rule:${{dF}_{\theta|r}(\theta)} = \frac{{f_{c}\left( {r,\theta} \right)}{{dF}_{\theta}(\theta)}}{\int{{f_{c}\left( {r,\theta} \right)}{{dF}_{\theta}(\theta)}}}$

[0061] where f_(c)(•, θ) is the probability density associated toP_(c)(•, θ). The customer's cost is independent of the quantity q chosenby the provider in stage 4, so that Equation (14) becomes$r \in {\arg \quad {\min\limits_{r}{K\left( {p_{0},p_{1},r,\theta} \right)}}}$

[0062] It can be readily seen that this is a newsvendor-like problem andthat the optimal quantity r* of options is a solution of the equation$\begin{matrix}{r^{*} = {F_{D|\theta}^{- 1}\left( {1 + \frac{p_{0}}{p_{1} - p}} \middle| \theta \right)}} & (16)\end{matrix}$

[0063] Therefore, the customer has a unique, pure equilibrium strategyr* (p₀, p₁, θ) given by Equation (16); r* is independent on theprovider's choice of q in the following stage of the game. To computethe provider's equilibrium strategy, we rewrite Equation (15). Theprovider observes r*, and maximizes his or her expected profitconditionally on the information that θ is in the set $\begin{matrix}{T_{r^{*}} = \left\{ {\left. {\theta \in \Theta} \middle| {\arg \quad {\min\limits_{r}{K\left( {p_{0},p_{1},r,\theta} \right)}}} \right. = r^{*}} \right\}} & (17)\end{matrix}$

[0064] And the provider's optimal reply is a pure strategy q* given by$\arg \quad {\max\limits_{r}{E\left( {{{p_{0}r^{*}} + {\left( {p_{1} - p} \right)\left( {D\bigwedge r^{*}} \right)} + {\Pi \left( {q,\left. \theta \middle| T_{r^{*}} \right.} \right)}},} \right.}}$

[0065] or, equivalently,$\arg \quad {\max\limits_{q}{{E\left( {\Pi \left( {q,\theta} \right)} \middle| T_{r^{*}} \right)}.}}$

[0066] Having found the optimal strategy of the subgame, we use Equation(22), below, to determine the optimal pricing strategy (p₀, p₁) of theprovider in stage 2. $\begin{matrix}{V = {\max\limits_{p_{0},p_{1}}{E\left\lbrack {{p_{0}\left( r^{*} \right)} + {\left( {p_{1} - p} \right)\left( {D\bigwedge r^{*}} \right)} + {E\left( {\Pi \left( {q^{*},\theta} \right)} \middle| T_{r^{*}} \right)}} \right\rbrack}}} & (18)\end{matrix}$

[0067] We notice that the inequalities $\begin{matrix}{{E\left( {{p_{0}r^{*}} + {\left( {p_{1} - p} \right)\left( {D\bigwedge r^{*}} \right)}} \right)} \leq 0} & (19) \\{{E\left( {E\left( {\Pi \left( {q^{*},\theta} \right)} \middle| T_{r^{*}} \right)} \right)} \leq {E\left( {\max\limits_{q}{\Pi \left( {q,\theta} \right)}} \right)}} & (20)\end{matrix}$

[0068] hold for all p₀, p₁. With regards to the former inequality, weuse the inequality t,0185

[0069] Using Equations (23), below, and (10), above, we have

E(p ₀ r*+(p ₁ −p)(DΛr*)+pE(D))≦pE(D)

[0070] and the result follows. Inequality (20) follows from${\left. {E\left( {{E{\prod\left( {q^{*},\theta} \right)}}T_{r^{*}}} \right)} \right) = {{E\left( {\max\limits_{q}{E\left( {{\prod\left( {q,\theta} \right)}T_{r^{*}}} \right)}} \right)} \leq {E\left( {\max\limits_{q}{\prod\left( {q,\theta} \right)}} \right)}}},$

[0071] where the last inequality is Equation (6). The previousinequalities yield an immediate upper bound for the maximum expectedpayoff of the provider (Equation (18)):$V \leq {E\left( {{\max\limits_{q}\quad {E\left( {\prod\left( {q,\theta} \right)} \right)}} = {V_{FB}.}} \right.}$

[0072] We now show that this a payoff arbitrarily close to this upperbound is actually attained under the assumptions of the theorem.

[0073] LEMMA 4: If p₀ satisfies Equation (9) then

[0074] 1. The pure equilibrium strategy of the provider is given by

q*(r* (θ))=r*.

[0075] 2. Equation (23) holds as an equality.

[0076] PROOF: The optimal strategy is given by $\begin{matrix}{{q^{*}\left( {p_{0},p_{1},r^{*}} \right)} = {\arg \quad \underset{q}{\max \quad}\quad {E\left( {{\prod\left( {p_{0},p_{1},q,\theta} \right)}T_{r^{*}}} \right)}}} \\{= {{\arg \quad {\max\limits_{q}\quad {\left( {s - c} \right)q}}} - {\left( {c^{\prime} - s} \right){E\left( {{Dq}T_{r^{*}}} \right)}}}}\end{matrix}$

[0077] by substituting p₀ from Equation (20), we obtain $\begin{matrix}{{{\arg {\quad \quad}{\max\limits_{q}{\left( {s - c} \right)q}}} - {\left( {c^{\prime} - s} \right){E\left( {{Dq}T_{r^{*}}} \right)}}} = {{\arg \quad {\min\limits_{r}\quad {p_{0}r}}} +}} \\{{\left( {p_{1} - p} \right){E\left( {{Dr}T_{r^{*}}} \right)}}} \\{= {\arg \quad {\min\limits_{r}\quad {E\left( {{K\left( {p_{0},p_{1},r,\theta} \right)}T_{r^{*}}} \right)}}}}\end{matrix}$

[0078] By Equation (17) we have${{r^{*}\left( {p_{0},p_{1},\theta} \right)} = {\arg \quad {\min\limits_{r}{K\left( {p_{0},p_{1},r,\theta} \right)}}}},$

[0079] for all θ ε T_(r*), from which we have${\arg \quad {\min\limits_{r}{E\left( {{K\left( {p_{0},p_{1},r,\theta} \right)}T_{r^{*}}} \right)}}} = {r^{*}.}$

[0080] Given the value r* from the customer, the provider knows that θ εT_(r*). The probability distribution dF_(θ|r*). is supported by the setT_(r*); i.e., P_(θ|r*)(T_(r*))=1. It follows that $\begin{matrix}{V_{FB} = {E\left( {\max\limits_{q}{\prod\left( {q,\theta} \right)}} \right)}} \\{= {E\left( {E\left( {{\max\limits_{q}{\prod\left( {q,\theta} \right)}}T_{r^{*}}} \right)} \right)}} \\{= {{E\left( {\max\limits_{q}\quad {E\left( {{\prod\left( {q,\theta} \right)}T_{r^{*}}} \right)}} \right)}.}}\end{matrix}$

[0081] The last equality follows from the observation that${\max\limits_{q}{\prod\left( {q,\theta} \right)}} = {\prod\left( {r^{*},\theta} \right)}$

[0082] for all θ ε T_(r*).

[0083] LEMMA 5: If p₀, p₁ satisfy Equations (8), (9), we have

|E(p ₀ r*+(p ₁ −p)(DΛr*))|<ε.

[0084] PROOF: We first observe that, from Equations (7),(17), we haver*(p₀, p₁, θ)={circumflex over (q)}(θ). Choose p₀, p₁ such that$0 < {p - {p1}} < {\frac{\varepsilon}{E\left( {\hat{q}(\theta)} \right)}.}$

[0085] We have${{E\left( {p_{0},{r^{*} + {\left( {p_{1} - p} \right)\left( {Dr^{*}} \right)}}} \right)}} = {{\left( {p - p_{1}} \right){{E\left( {{{\frac{c - s}{c^{\prime} - s}r^{*}} - D}r^{*}} \right)}}} < {\left( {p - p_{1}} \right){E\left( {\hat{q}(\theta)} \right)}} < \varepsilon}$

[0086] From application of the previous lemmas to Equation (18) theresult of Theorem 2 follows.

[0087] The result states that, under the prescribed pricing scheme, thecustomer can attain an expected profit that is arbitrarily close fromthe maximum possible attainable profit.

[0088] There is an intuitive explanation for the above result. Seeingprices p₀, p₁, p, the customer chooses a capacity r that minimizes hisor her expected cost. The problem the customer faces is${\min\limits_{r}\quad {p_{0}r}} + {p_{1}\quad {E\left( {Dr} \right)}} + {{pE}\left( {D - r} \right)}^{+}$

[0089] It can be readily seen that this is a news vendor-like problemand that the optimal quantity r* of options is such that

P ₀+(p ₁ −p)(1−F _(D|θ)(r*|θ))=0

[0090] or, after substitution of (p₀ using Equation (9),

s−c+(c′−s)(1−F _(D|θ)(r*|θ))=0

[0091] Therefore, r* is equal to the optimal capacity that the providerwould choose in a linear pricing contract if the provider knew the typeθ of the customer.

[0092] Another prescription of Theorem 2 is that the optimal initialcapacity investment should be equal to the discount threshold r*purchased by the customer. This is a consequence of the particularchoice of the parameters p₀, p₁. For arbitrary price parameters, theoptimal capacity investment is in general different than r*.

[0093] A closely related result states that the new schedule can be usedto obtain expected allocations that are Pareto-superior compared to theoriginal pricing.

[0094] COROLLARY 3: Let V₁ be the provider expected profit defined inEquation (5), and let C₁=V₁ be the customer expected cost in the basiccontract. Let${p_{1} \in \left( {0,p} \right)},{p_{0}\frac{c - s}{c^{\prime} - s}\left( {p - p_{1}} \right)},$

[0095] and let V(p₀, p₁), C(p₀, p₁) the expected profit (cost) of theprovider (customer) under the flexible discount contract.

[0096] 1. The utilitarian welfare of the provider and customer is equalto

V((p ₀ , p ₁)−C(p ₀ , p ₁))=V _(FB) −pE(D).

[0097] 2. The expected customer's cost is linearly increasing as afunction of

[0098] 3. $\begin{matrix}{{{Let}\quad \delta^{*}} = {{{\frac{V_{FB} - V_{1}}{E\left( {D{{\hat{q}(\theta)} + {\left( {1 - \hat{f}} \right){\hat{q}(\theta)}}}} \right)}.\quad {If}}\quad p_{1}} > {p - \delta^{*}}}} & (21)\end{matrix}$

[0099] the resulting allocation is Pareto improving upon the originalallocation:

V(p ₀ , p ₁)>V ₁

C(p ₀ , p ₁)<C ₁

[0100]FIG. 4 shows the expected allocations under the linear andflexible discount pricing. The allocation ξ₀=(V₁, pE(D)) corresponds tothe linear pricing contract. Under the flexible discount contract, acontinuum of allocations can be achieved within the contract, i.e.,without the need of ex post monetary transfer. If the contractparameters are parameterized

δε(0, (V _(FB) −V ₁)(E(DΛ{circumflex over (q)}(θ)+(1−{circumflex over(f)}){circumflex over (q)}(θ)))⁻¹

p ₁ =p−δ

p ₀=(1−{circumflex over (f)})δ

[0101] then the set of Pareto-improving allocation is given by thefollowing curve:

ξ_(δ)=(V _(FB)(1−δ)+V ₁δ), pE(D)−(V _(FB) −V ₁)δ)

The Case of Normal Demand

[0102] The result is illustrated in the important special case of normaldemand. The customer type is given by the pair θ=(μ, σ), and writtenμ(θ), σ(θ). The customer has a prior distribution P on Θ.

[0103] Under perfect knowledge of the customer's type, the optimalcapacity investment is expressed by Equation (7):

{circumflex over (q)}(θ)=μ(θ)+σ(θ)Φ−1({circumflex over (f)}),

[0104] where (Φ(•) is the cumulative distribution function a standardnormal random variable.

[0105] To compute the expected profit under perfect knowledge, we definea, b as follows: $\begin{matrix}{a = {p - c}} \\{b = {\left( {c^{\prime} - s} \right){\int_{\Phi^{- 1}{(\hat{f})}}^{\infty}{x\quad {{\Phi (x)}}}}}}\end{matrix}$

[0106] Note that both a and b are positive. $\begin{matrix}{{\left. {{E\left( {D{\hat{q}(\theta)}} \right)} = {\int_{- \infty}^{\Phi^{- 1}{(\hat{f})}}{\left( {{\mu (\theta)} + {{\sigma (\theta)}x}} \right)\quad {\Phi}}}} \right)(x)} +} \\{{\left( {{\mu (\theta)} + {{\sigma (\theta)}{\Phi^{- 1}\left( \hat{f} \right)}}} \right)\left( {1 - \hat{f}} \right)}} \\{= {{\mu (\theta)} + {{\sigma (\theta)}\left( {{\Phi^{- 1}\left( \hat{f} \right)} - {\int_{\Phi^{- 1}{(\hat{f})}}^{\infty}{x\quad {{\Phi (x)}}}}} \right)}}}\end{matrix}$

[0107] Applying this formula we get $\begin{matrix}\begin{matrix}{{\max\limits_{q}\quad {E\left( {{\pi \left( {q,D} \right)}\theta} \right)}} = {{{pE}(D)} - {c\hat{q}} - {c^{\prime}{E\left( {D - {\hat{q}(\theta)}} \right)}^{+}} +}} \\{{{sE}\left( {{\hat{q}(\theta)} - D} \right)}^{+}} \\{= {{\left( {s - c} \right){\hat{q}(\theta)}} + {\left( {c^{\prime} - s} \right){E\left( {D{\hat{q}(\theta)}} \right)}} -}} \\{{\left( {c^{\prime} - p} \right){E(D)}}} \\{= {{a\quad {\mu (\theta)}} - {b\quad {\sigma (\theta)}}}}\end{matrix} & (22) \\\begin{matrix}{{{{Since}\quad V_{FB}} = {E\left( {\max\limits_{q}\quad {E\left( {{\pi \left( {q,D} \right)}\theta} \right)}} \right)}},{{we}\quad {have}}} \\{{VFB} = {{{aE}(\mu)} - {{bE}(\sigma)}}}\end{matrix} & (23)\end{matrix}$

[0108] Moreover, the value of V₁ can be computed by noticing that thedistribution of D in the absence of information on types is stillnormally distributed, with mean equal to Eμ(θ) and standard deviationequal to (Eσ²(θ))^(1/2). From Equation (12) we immediately obtain

V ₁ =aE(μ)−b(Eσ ²)^(1/2).

[0109] The value of information in the case of normally distributeddemand admits a simple formula, which is independent on the priordistribution on the mean, but depends on the first two moments of thestandard deviation with respect to the prior measure P on thecustomers'types.

V _(FB) −V ₁ =b((E((σ²))^(1/2) −E(σ)).

[0110] Let us defined = (2 + f̂)E(μ) + (2(1 + f̂)Φ⁻¹)(f̂) − ∫_(Φ⁻¹(f̂))^(∞)x  Φ(x))E(σ).

[0111] The lower bound for Pareto-improving prices p, is given by${p - \frac{V_{FB} - V_{1}}{E\left( {D{{\hat{q}(\theta)} + {\left( {+ \hat{f}} \right){\hat{q}(\theta)}}}} \right)}} = {p - {\frac{b}{d}{\left( {\left( {E\left( \sigma^{2} \right)} \right)^{1/2} - {E(\sigma)}} \right).}}}$

[0112] The properties of a class of contracts that are beingincreasingly adopted in the utility industry were investigated todetermine the monetary transfers between a provider of the service and acustomer. In these contracts, the provider faces an initial capacityinvestment decision in the face of uncertain demand. The contractenables the provider to obtain from the customer the information neededfor optimal ex ante capacity planning. The resulting utilitarian welfareis first-best, and can be achieved for any users' type distribution anddemand distribution function. Furthermore, the surplus can be allocatedin any proportion among customer and provider without the need ofout-of-contract monetary transfers.

[0113] The flexible discount contract described above bears a similarityto signaling models (M. Spence, “Job market signaling”, The QuarterlyJournal of Economics, 87(3):355-374, 1973) and to models of preplaycommunication, or “cheap talk”(V. P. Crawford and J. Sobel, “Stategicinformation transmission”, Econometrica, 50(6):1431-1451, 1982). To makethe connection clear, the last two stages of the contract areconsidered, in which the customer first chooses a threshold level andthen the provider makes a capacity planning decision. In this subgame,the informed party (the customer) moves first, and his or her actionreveals information about his or her type to the uninformed party (theprovider), who uses it when he or she has to provide for capacity. Theprovider does not obtain full disclosure of the customer's type; yet,the knowledge of the capacity threshold selected by the customer issufficient to make an optimal capacity planning decision. This subgameis therefore similar to the standard signaling setting, in that theinformed party moves first by sending a costly signal. On the otherhand, it is similar to models of preplay communication, in that thepayoff of the informed player is not a direct function of the player'stype.

[0114]FIG. 5 is a flowchart showing the overall process according to theinvention. The process begins in function block 51 where the customerselects a capacity discount threshold. During the providing of servicesunits (SUs), the provider 10 monitors in function block 52 the load ofthe customer with the load monitor 112 (see FIG. 1). A determination ismade in decision block 53 as to whether the customer demand exceeds theselected capacity discount threshold. If not, the pricing and billingcomponent 114 (FIG. 1) generates a bill to the customer at the baseprice rate in function block 54. On the other hand, if the customerdemand exceeds the selected capacity discount threshold, then thepricing and billing component 114 first calculates the peak price forthe services received in function block 55 and then generates a bill tothe customer at the peak price rate in function block 56.

[0115]FIG. 6 is the flowchart of the monitoring process performed by theload monitor 112 (FIG. 1) in function block 52 (FIG. 5). Time t isinitialized to zero in function block 61 at the beginning of theprocess. Then a processing loop is entered at the beginning of which theload time period T is incremented by one in function block 62. Ameasurement is made of SU(t) in function block 63. A determination ismade in decision block 64 as to whether t=T and, if not, the processloops back to function block 62; otherwise, the measured load timeseries is stored in repository 113 (FIG. 1) before the processterminates.

[0116]FIG. 7 is a flowchart of the computation process of the pricingand billing component 114 (FIG. 1) performed in function block 55 (FIG.5). The process begins by initializing the Charge to p₀r and time t tozero in function block 71. The process then enters a processing loopwhich begins by computing the Charge as

Charge+p ₁ min {SU(t), r}+p max {SU(t)−r, 0}

[0117] in function block 72. A determination is made in decision block73 as to whether t=T and, if not, the process loops back to functionblock 72; otherwise, the bill is generated based on the computation andthe process ends.

[0118] While the invention has been described in terms of a singlepreferred embodiment, those skilled in the art will recognize that theinvention can be practiced with modification within the spirit and scopeof the appended claims.

Having thus described my invention, what I claim as new and desire tosecure by Letters Patent is as follows:
 1. A method for design ofpricing schedules in utility contracts comprising the steps of: before acontract starting date, selecting by a customer a capacity discountthreshold, said capacity discount threshold being a prespecified rate ofprovisioning by a provider of standardized services, a price paid by thecustomer to the provider for the standardized services beingproportional to the selected threshold; during a term of the contract,measuring by the provider demand by the customer of the standardizedservices; and if demand rate by the customer of the standardized servicestays below the selected threshold, paying by the customer a base priceper unit of standardized services received, but if the instantaneousdemand rate by the customer of standardized service exceeds the selectedthreshold, paying by the customer a peak price per unit of standardizedservices received, which peak price is greater than the base price. 2.The method of claim 1, wherein a contract interval is divided into Nsampling intervals of equal length, for each sampling interval n=1, . .. ,N, the step of measuring by the provider measures a number X_(n) ofservice units (SUs) provided to a customer.
 3. The method of claim 2,wherein before a starting date of a contract, choosing by the provider aresource capacity q, where q is defined as a maximum number of SUs thatcan be served during a sampling period, wherein a unit cost of theresource capacity q is c per sampling period, and wherein if demandduring a sampling interval exceeds the resource capacity q, the providercan serve the demand by incurring a unit cost equal to c′ which isgreater than c.
 4. The method of claim 3, wherein the provider selects apositive parameter epsilon, with epsilon<p, and sets parametersp0=(c−s)/(c′−s)*epsilon and p1=p-epsilon, and wherein the step ofselecting by a customer a capacity discount threshold the customerreserves ex ante a discount threshold r, for which the customer pays aunit price Np0, and wherein during a sampling interval, paying by thecustomer a discounted unit price P1, if load does not exceed r andpaying by the customer a full price p if the load exceed r.
 5. Themethod of claim 3, wherein the provider selects a positive parameter ε,with ε<p, and sets parameters p0=(c−s)/(c′−s)*ε and p1=p−ε, and whereinthe step of selecting by a customer a capacity discount threshold thecustomer reserves ex ante a capacity r, for which the customer pays aunit price N(p0+p1), and wherein during a sampling interval, paying bythe customer a discounted unity price p1, if load does not exceed r andpaying by the customer a full price p if the load exceed r.
 6. A systemfor facilitating the design of pricing schedules in utility contractscomprising: a provider of standardized services to a plurality ofcustomers wherein, before a contract starting date, each of theplurality of customers selects a capacity discount threshold, saidcapacity discount threshold being a prespecified rate of provisioning bythe provider of standardized services, a price paid by the customer tothe provider for the standardized services being proportional to theselected threshold, an allocated capacity by the provider equal to thesum of the capacity discount threshold selected by the customers; a loadmonitor at the provider for monitoring, during terms of contracts withsaid plurality of customers, demands by each customer of said pluralityof customers of the standardized services provided by the provider; anda pricing and billing component at the provider and responsive tomonitored demands by each customer of said plurality of customers todetermine if demand rate by a customer of the standardized service staysbelow the threshold selected by the customer, and if so, billing thecustomer a base price per unit of standardized services received, but ifthe instantaneous demand rate by the customer of standardized serviceexceeds the threshold selected by the customer, billing the customer apeak price per unit of standardized services received, which peak priceis greater than the base price.
 7. The method of claim 4, wherein theprovider selects a positive parameter ε, with ε<p, and sets parametersp0=(c−s)/(c′−s)*ε and p1=p−ε, and wherein the step of allocating by theprovider a capacity q the provider allocates q, equal to the capacitythreshold reserved ex ante by the customer.